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8.21 The Physics of Energy
Fall 2009
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
8.21 Lecture 6
Quantum Mechanics I
September 21, 2009
8.21 Lecture 6: Quantum Mechanics I
1 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
The QUANTUM WORLD is a STRANGE PLACE!
Position of an object is not well-defined
Objects can tunnel through barriers
Energy, momentum, etc. become discretized
But quantum physics is crucial for energy processes
• Discrete quantum states ⇒ entropy ⇒ thermo ⇒ limits to efficiency
• Nuclear processes: fission + fusion depend on tunneling
• Absorption of light by matter (atmosphere, photovoltaics, etc.):
depends on discrete quantum spectrum
This lecture: QM rapid immersion
8.21 Lecture 6: Quantum Mechanics I
2 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Quantum wavefunctions
Classically, particles have position x, momentum p
r
x
p
In quantum mechanics, particles are described by wavefunctions
ψ(x)
Wavefunction obeys (time-dependent) Schrödinger wave equation
∂
~2 ∂ 2
∂2
∂2
i~ ψ(x, t) = −
ψ(x,
t)
+
ψ(x,
t)
+
ψ(x,
t)
+
V(x)ψ(x,
t)
∂t
2m ∂x2
∂y2
∂z2
Why do we believe this wacky notion?
• Vast range of experiments over last 100 years
• Foundation of most of modern physics.
8.21 Lecture 6: Quantum Mechanics I
3 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
2-slit experiment. Shoot particles through one or two slits at screen
Destructive + constructive interference ⇒ particles are waves
8.21 Lecture 6: Quantum Mechanics I
4 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
2-slit experiment. Shoot particles through one or two slits at screen
Destructive + constructive interference ⇒ particles are waves
8.21 Lecture 6: Quantum Mechanics I
4 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Many phenomena described by waves
frequency significance
1 ∂2
E
c2 ∂t2
light (EM)
sound (pressure)
QM (wavefunction)
= ∇2 E
color
1 ∂
p
v2s ∂t2
= ∇2 p
pitch
∂
∂t ψ
i~
2
2m ∇ ψ
energy
2
=
Wave equation is linear:
ψ1 (x, t) and ψ2 (x, t) solve ⇒ linear combination aψ1 (x, t) + bψ2 (x, t) solves
+
=
Exhibit constructive and destructive interference for phases in/out of sync.
8.21 Lecture 6: Quantum Mechanics I
5 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
2
2
∂
∂
Violin string: ρ ∂t
2 Y(x, t) = T ∂x2 Y(x, t)
Yn = cos(nω1 t) sin(n · πL x) ωn = nω1
Y3 = cos(3ω1 t) sin(3 · πL x) ω3 = 3ω1
Y2 = cos(2ω1 t) sin(2 · πL x) ω2 = 2ω
q1
Y1 = cos(ω1 t) sin( πL x)
ω1 = πL ρT
• Modes–higher harmonics (n = 2, 4, 8, . . . up by octaves)
• ωn ∼ n (2∂/∂t’s, 2∂/∂x’s)
• Pluck string – get superposition (linear combination) of modes
Solutions: sine modes
2
2
∂
~ ∂
ψ(x, t)
ψ(x, t) = − 2m
Quantum particle in a 1D box: i~ ∂t
∂x2
Sine modes again
ψn = e−iEn t/~ sin(n · πL x) En = n2 ~ω1
···
···
ψ2 = e−iE2 t/~ sin(2 · πL x) E2 = 4~ω1
2 2
ψ1 = e−iE1 t/~ sin( πL x)
E1 = ~ω1 = π2mL~2
• Each mode – state of fixed energy
• En ∼ n2 (1∂/∂t, 2∂/∂x’s)
• General state – superposition (linear combination) of modes
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Quantum wavefunctions are complex – Review of complex numbers
Define i2 = −1
Complex number: z = x + iy
Often write z = reiθ = r(cos θ + i sin θ)
r = magnitude, θ = phase
Useful properties of complex numbers:
Addition: (x + iy) + (a + ib) = (x + a) + i(y + b)
Multiplication: (x + iy) × (a + ib) = (xa − yb) + i(ya + xb)
(reiθ )(seiψ ) = rsei(θ+ψ)
Complex conjugation: z̄ = z∗ = x − iy
p
Norm: |z| = x2 + y2 , |reiθ | = r, |z|2 = zz̄ = r2
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
General quantum wavefunction
Quantum particle has “energy basis” spatial wavefunctions ψi (x)
ψi (x) have fixed energies Ei
General (time-dependent) state is superposition
ψ(x, t) = a1 e−iE1 t/~ ψ1 (x) + a2 e−iE2 t/~ ψ2 (x) + · · ·
For macroscopic (classical) systems, combine many quantum states
• Destructive interference outside small region ⇒ classical localization
• Wavefunction nonzero through classical barriers ⇒ tunneling
• For micro systems (e.g. atoms) individual quantum states relevant.
8.21 Lecture 6: Quantum Mechanics I
8 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Rules of Quantum Mechanics: 4 Axioms
Energy in quantum mechanics
Axiom 1: Any finite/physical quantum system has a discrete set of
“energy basis states”, which we denote s1 , s2 , . . . , sN .
These states have values of energy E1 , E2 , . . . , EN .
Example: hydrogen atom
0
Example: semiconductor
···
E11−26 ∼
= 0 /9 (3s,p,d)
E3−10 ∼
= 0 /4 (2s,p)
[0 ∼
= -13.6 eV]
E1,2 ∼
= 0
(1s)
bands R
@
?
6band gap
• important for photovoltaics
• Values of E: “spectrum”
• Physicists’ job: compute spectrum of physical systems
— Often deal with ∞ state approximation
8.21 Lecture 6: Quantum Mechanics I
9 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Simplest quantum system: “Qubit” = 2-state system (electron spin)
Earth spins
Electron spins
or
Lz = ± 21 ~
Classically any ω seems ok
⇒ any L, Erot
Electron in magnetic field B = Bẑ
E = −B · µ = µ̃BLz
B
6
6
6
6
6
6
6
|+i 6
|−i 6
E± = ±µ̃B~/2
• 2 states
•~∼
= 1.0546 × 10−34 Js
–fundamental quantum unit
Confirmed by experiment
N
S
8.21 Lecture 6: Quantum Mechanics I
|+i
|−i
10 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Axiom 2: The state of a quantum system at any point in time is a linear
combination (“quantum superposition”) of basis states
|si = z1 |s1 i + z2 |s2 i + · · · + zn |sn i
• Can think of like a vector: r = xî + yĵ + zk̂
• Convention: unit normalization |z1 |2 + |z2 |2 + · · · + |zn |2 = 1
What does a quantum superposition mean?
Axiom 3: If you measure the system’s energy (assume Ei distinct)
probability(E = Ei ) = |zi |2 , after measurement state ⇒ si
N
Example:
√1 |+i
2
+
√1 |−i
2
S
|+i, prob = 1/2
|−i, prob = 1/2
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Puzzle: If measurements give random results, how is E conserved?
Example: 2 separated electrons in B field, total E = 0
B
B
6
6
6
6
6
6
6
|+i 6
|−i 6
state 1: |+−i
6
6
6
6
6
6
6
|−i 6
|+i 6
state 2: |−+i
Both states: E = E+ + E− = 0
Assume system in state
√1 |+−i
2
+
√1 |−+i
2
Measure spin of first particle
50%: Particle 1 in state |+i: system in state 1, E1 = E+ , E2 = E−
50%: Particle 1 in state |−i: system in state 2, E1 = E− , E2 = E+
BUT TOTAL ENERGY IS CONSERVED!
8.21 Lecture 6: Quantum Mechanics I
12 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Time Dependence
Axiom 4: If at time t0 a state |s(t0 )i has definite energy E
then at time t the state is
|s(t)i = e−iE(t−t0 )/~ |s(t0 )i
⇒
[
]
Time evolution is linear in |si, so if
|s(t0 )i = z1 |s1 i + · · · + zn |sn i
then |s(t)i = z1 e−iE1 (t−t0 )/~ |s1 i + · · · + zn e−iEn (t−t0 )/~ |sn i
Note: only phase changes for definite E state!

z1


|s(t0 )i =  ... 
zn

Matrix notation
⇒ Schrödinger equation
d
dt |s(t)i
d
dt |s(t)i
= − ~i E|s(t)i
E1
 0

0
E2
H=

0
···
..
0
0
0 



.
En
= − ~i H|s(t)i
8.21 Lecture 6: Quantum Mechanics I
13 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
SUMMARY: axioms of quantum mechanics
1. Any physical system has a discrete set of E basis states |si i
2. General state is linear combination z1 |s1 i + · · · + zn |sn i
3. Measurement: probability = |zi |2 that state → |si i
4. Time development linear, |si i → e−iEi (t−t0 )/~ |si i
• All of QM, QFT essentially elaboration on principles 1-4 + symmetry,
developing tools for calculations in particular cases
• Often work in another basis (not E basis)
• A primary problem: given system, determine spectrum
8.21 Lecture 6: Quantum Mechanics I
14 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Particle states described by wavefunctions
|ψ(x)|2
ψ(x)
AK
A
Like superposition of states in fixed positions, |ψ(x)| = prob. @ x
2
Think of as limit of discrete “position basis”
x1 x2 x3 x4 x5
|ψi =
P
j
P
j
ψj |xj i
|ψj |2 = 1
→
→
→
ψ(x)
R
|ψ(x)|2 dx = 1
8.21 Lecture 6: Quantum Mechanics I
15 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
What are energy basis states (|si’s) for free particle?
Translate ⇒ same Energy: ψ(x + δ) = eiθδ ψ(x)
• Plane wave states ψp (x) = eipx/~
p = momentum!
• Matches experimental observation (Davisson-Germer, 1927):
de Broglie wavelength of matter λ = h/p (eipx/~ = e2πix/λ )
• Energy: E =
p2
2m
Schrödinger equation:
Ep ψp (x) =
p2
~2 ∂ 2
ψp (x) = −
ψp (x) = Hψp (x)
2m
2m ∂x2
2
2
∂
~ ∂
Time-dependent Schrödinger eq.: i~ ∂t
ψ(x, t) = Hψ = − 2m
ψ(x, t)
∂x2
8.21 Lecture 6: Quantum Mechanics I
16 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Back to particle in a (1D) box
E6 = 36
E5 = 25
Time-independent Schrödinger equation:
Hψ(x) = −
~2 ∂ 2
ψ(x) = Eψ(x)
2m ∂x2
E4 = 16
E3 = 9
Boundary Conditions: ψ(0) = ψ(L) = 0
Solution: Combination of eipx/~ , e−ipx/~
Energy basis states |ni: ψn = sin πnx/L
E2 = 4
E1 = Spectrum ( = ~2 π 2 n2 /2mL2 )
2 2 2
En =
~ π n
2mL2
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Even more useful model: 1D Simple Harmonic Oscillator (SHO)
..
.
E5 = 5 12 ~ω
V(x) = 21 kx2
E4 = 4 12 ~ω
~2 ∂ 2
1 2
Hψ(x) = −
kx
ψ(x) = Eψ(x)
+
2m ∂x2
2
E3 = 3 12 ~ω
E2 = 2 12 ~ω
Boundary Conditions: ψ(|x| → ∞) = 0
p
Solution: [ω = k/m]
|0i :
|1i :
|2i :
···
E1 = 1 12 ~ω
mω 2
ψ0 = C0 e− 2~ x
mω 2
ψ1 = C1 xe− 2~ x
2
2
− mω
2~ x
ψ2 = C2 ( 2mω
~ x − 1)e
E0 = 12 ~ω
Spectrum (En = (n + 1/2)~ω)
[Analytic solution; many approaches, one in notes]
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
Hydrogen-like atom (now in 3D, assume mp me )
0
E3
E2
···
∼
= −13.6 eV/9
∼
= −13.6 eV/4
[wikimedia]
~2 2
e2
Hψ(x) = − ∇ −
ψ(x) = Eψ(x)
2m
4π0 r
Solutions:
ψ1s = C1s × (a0 )−3/2 e−r/a0
ψ2s = C2s × (a0 )−3/2 e−r/a0 (1 − r/2a
0)
ψ2p = C2p × (a0 )−3/2 e−r/a0 x,y,z
r
···
~2
where Bohr radius is a0 = me
2 4π0 ≈ 0.52Å
E1 ∼
= −13.6 eV
Spectrum
2
∼ −13.62 eV )
(En = 4π−e
2 =
n
0 a0 n
8.21 Lecture 6: Quantum Mechanics I
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Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
How does quantum state ⇒ classical physics?
Consider particle in potential
Sum of energy basis states ⇒ “localized wave packet”
Time evolution under Schrödinger equation ψ̇(t) =
−i
~ Hψ(t)
⇒ Packet follows classical laws
Z
hxiψ(t) =
dx x|ψ(x, t)|2
Z
Define
X
∂
hpiψ(t) =
ψ̄p ψp p = −i~ dx ψ̄(x, t) ψ(x, t) .
∂x
p
d
1
hxiψ(t) =
hpiψ(t)
dt
m
Can show
d2
d
∂
m 2 hxiψ(t) =
hpiψ(t) = −h V(x)iψ(t)
dt
dt
∂x
8.21 Lecture 6: Quantum Mechanics I
20 / 21
Quantum wavefunctions
Rules of Quantum Mechanics
Quantum Mechanics of Particles
SUMMARY of QM
• Quantum particles described by wavefunction
• Any quantum system: basis of states w/ fixed energy (A1)
General state linear combination (superposition) of energy basis (A2)
• Quantum particles: for V = 0, eipx/~ has momentum p, E = p2 /2m
h 2 2
i
~ ∂
• Energy basis states with potential V : Hψ = − 2m
+
V
ψ = Eψ
2
∂x
• Box spectrum: En = ~2 π 2 n2 /2mL2 , n = 1, 2, . . .
• SHO spectrum: En = (n + 1/2)~ω, n = 0, 1, . . .
• Time-dependent Schrödinger eq. ψ̇(t) =
−i
~ Hψ(t)
[E ∝ frequency] (A4)
8.21 Lecture 6: Quantum Mechanics I
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